Geodesic
$\phi({\rm Ric})$-vector fields in Riemannian spaces
Archivum mathematicum, Tome 44 (2008) no. 5, pp. 385-390 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper we study vector fields in Riemannian spaces, which satisfy $\nabla \varphi =\mu $, ${\textbf{Ric}}$, $\mu =\mbox {const.}$ We investigate the properties of these fields and the conditions of their coexistence with concircular vector fields. It is shown that in Riemannian spaces, noncollinear concircular and $\varphi (\mbox {\textbf{Ric}})$-vector fields cannot exist simultaneously. It was found that Riemannian spaces with $\varphi (\mbox {\textbf{Ric}})$-vector fields of constant length have constant scalar curvature. The conditions for the existence of $\varphi (\mbox {\textbf{Ric}})$-vector fields in symmetric spaces are given.
In this paper we study vector fields in Riemannian spaces, which satisfy $\nabla \varphi =\mu $, ${\textbf{Ric}}$, $\mu =\mbox {const.}$ We investigate the properties of these fields and the conditions of their coexistence with concircular vector fields. It is shown that in Riemannian spaces, noncollinear concircular and $\varphi (\mbox {\textbf{Ric}})$-vector fields cannot exist simultaneously. It was found that Riemannian spaces with $\varphi (\mbox {\textbf{Ric}})$-vector fields of constant length have constant scalar curvature. The conditions for the existence of $\varphi (\mbox {\textbf{Ric}})$-vector fields in symmetric spaces are given.
Classification : 53B05, 53B30
Keywords: special vector field; pseudo-Riemannian spaces; Riemannian spaces; symmetric spaces; Kasner metric 
@article{ARM_2008_44_5_a4,
     author = {Hinterleitner, Irena and Kiosak, Volodymyr A.},
     title = {$\phi({\rm Ric})$-vector fields in {Riemannian} spaces},
     journal = {Archivum mathematicum},
     pages = {385--390},
     year = {2008},
     volume = {44},
     number = {5},
     mrnumber = {2501574},
     zbl = {1212.53018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a4/}
}
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Hinterleitner, Irena; Kiosak, Volodymyr A. $\phi({\rm Ric})$-vector fields in Riemannian spaces. Archivum mathematicum, Tome 44 (2008) no. 5, pp. 385-390. http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a4/